OFFSET
1,2
COMMENTS
From Peter Bala, Dec 30 2023: (Start)
Denoted phi_2(n) in van der Kamp.
The number of solutions of the congruence x*y == 2 (mod n), 1 <= x, y <= n.
Can be regarded as a generalization of Euler's totient function phi(n) = Sum_{k = 1..n, gcd(k,n) = 1} gcd(k,n) since a(n) = Sum_{k = 1..n, gcd(k,n) divides 2} gcd(k,n). (End)
LINKS
N. Anghel, Heron triangles with constant area and perimeter, Rev. Roumaine Math. Pures Appl. 65 (2020), 4, 403-422.
Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, arXiv:1201.3139 [math.NT], 2012.
FORMULA
a(n) = phi(n) if n is odd; 2*phi(n) if n == 0 (mod 4); 2*phi(n) + phi(n/2) if n == 2 (mod 4).
From Ridouane Oudra, Oct 17 2021: (Start)
a(n) = Sum_{j = 1..n} gcd(n,j)*cos(4*Pi*j/n). (End)
From Peter Bala, Dec 30 2023: (Start)
a(n) = Sum_{d divides gcd(2,n)} d*phi(n/d), where phi(n) = A000010(n) denotes Euler's totient function.
Sum_{d divides n} a(d) = 2*n for n even, else equals n (van der Kamp, equation 26).
Dirichlet g.f.: zeta(s-1)*(1 + 2^(1-s))/zeta(s).
The Lambert series Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x*(1 + 4*x + x^2)/(1 - x^2)^2. See A022998.
Multiplicative with a(2) = 3, a(2^k) = 2^k for k >= 2 and a(p^k) = p*k - p^(k-1) for odd primes p.
If n divides m then a(n) divides 3*a(m). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 9/(2*Pi^2) = 0.455945... (A088245). - Amiram Eldar, Jan 18 2024
MAPLE
with(numtheory):
seq(add(d*phi(n/d), d in divisors(igcd(2, n))), n = 1..70); # Peter Bala, Dec 30 2023
MATHEMATICA
Table[If[OddQ[n], EulerPhi[n], If[Mod[n, 4]==0, 2EulerPhi[n], 2EulerPhi[n]+EulerPhi[n/2]]], {n, 68}] (* Stefano Spezia, Jul 30 2021 *)
PROG
(PARI) a(n) = if (n%2, eulerphi(n), if (n%4, 2*eulerphi(n) + eulerphi(n/2), 2*eulerphi(n)));
(Python)
from sympy import totient as phi
def a(n): return phi(n) if n%2 else 2*phi(n)+phi(n//2) if n%4 else 2*phi(n)
print([a(n) for n in range(1, 69)]) # Michael S. Branicky, Jul 30 2021
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Michel Marcus, Jul 30 2021
STATUS
approved